Let \(M\)  be a bounded linear operator on a Banach space \(E\) . The spectrum of \(M\) , which we denote by \(\sigma (M)\) , is the set of all complex numbers \(\lambda \)  for which \((\lambda I-M)\)  is not invertible, where \(I\)  denotes the identity operator. The spectral radius of \(M\)  is defined as \(r(M)=\sup \left\lbrace |\lambda |\colon \lambda \in \sigma (M)\right\rbrace \) . Let \(B_1(0)\)  be the unit ball in \(E\) . The operator \(M\)  is called compact if the image of \(B_1(0)\)  under \(M\)  is relatively compact. This is a rather strong condition. In fact, for the substitutions we treat in this work, the corresponding substitution operator is never compact; see [1]}. A weaker condition is quasi-compactness. An operator \(M\)  with spectral radius 1 is called quasi-compact, if there exists a compact operator \(N\)  and \(n\in \mathbb {N}\)  such that \(|M^n-N|^{ }_{\text{op}}<1\) , where \(|\cdot |^{ }_{\text{op}}\)  is the operator norm. We refer to [1]}, [3]} and references therein for details on such operators and their properties.
